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The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics.It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. [1] The terms path integral, curve integral, and curvilinear integral are also used; contour integral is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a ...
Since the series converges uniformly on the support of the integration path, we are allowed to exchange integration and summation. The series of the path integrals then collapses to a much simpler form because of the previous computation. So now the integral around C of every other term not in the form cz −1 is zero, and the integral is ...
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. [1][2][3] Contour integration is closely related to the calculus of residues, [4] a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are ...
Path integration. Path integration sums the vectors of distance and direction traveled from a start point to estimate current position, and so the path back to the start. Path integration is the method thought to be used by animals for dead reckoning.
The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇φ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as ...
Common integrals in quantum field theory. Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. [1]: 13–15 Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.
In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, [a] the other being differentiation. Integration was initially used to solve problems in mathematics and ...
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